Question

Simplify. $$\sec ^{2} x\left(1-\cos ^{2} x\right)$$

Short answer

\(\tan^2 x\)

Step-by-step solution

Recall Trigonometric Identity

Remember that one of the fundamental trigonometric identities is the Pythagorean identity, which is \(1 - \cos^2 x = \sin^2 x\). This identity will be used to simplify the given expression.

Apply Trigonometric Identity

Replace \(1 - \cos^2 x\) with \(\sin^2 x\) in the given expression. The new expression is \(\sec^2 x \cdot \sin^2 x\).

Simplify using the Definition of Secant

Recall the definition of secant, which is \(\frac{1}{\cos{x}}\). Substituting that in we get: \(\frac{1}{\cos^2 x} \cdot \sin^2 x\).

Apply Trigonometric Identity Again

Notice that the multiplication of \(\sin^2 x\) by \(\frac{1}{\cos^2 x}\) is equal to \(\tan^2 x\), since \(\tan x = \frac{\sin x}{\cos x}\). Therefore, the simplified form of the expression is \(\tan^2 x\).

Key concepts

Pythagorean Trigonometric Identity

Understanding the relationships between trigonometric functions is essential when it comes to simplifying expressions involving sine, cosine, and secant. A key player in this is the Pythagorean trigonometric identity, which is expressed as \(1 - \cos^2 x = \sin^2 x\). This equation is a restatement of the famous Pythagorean theorem within the context of a unit circle, where the sine and cosine represent the lengths of the legs of a right-angled triangle, and the radius of the circle (with a length of 1) represents the hypotenuse. By leveraging this identity, we can turn complex trigonometric expressions involving squares of sine and cosine functions into much simpler forms.

For instance, when faced with an expression such as \(\sec^2 x (1 - \cos^2 x)\), we can substitute the \(1 - \cos^2 x\) part using the Pythagorean identity. This transformation simplifies the problem, reducing it to a more manageable equation involving fewer trigonometric functions.

Secant Function

The secant function, denoted as \(\sec x\), is one of the lesser-known trigonometric functions but is incredibly useful in simplification. It is the reciprocal of the cosine function, which means \(\sec x = \frac{1}{\cos x}\). In a right-angled triangle, while cosine represents the ratio of the adjacent side to the hypotenuse, secant represents the ratio of the hypotenuse to the adjacent side. When \(\sec x\) is squared, it turns into \(\sec^2 x = \frac{1}{\cos^2 x}\), which can lead to further simplifications in trigonometric expressions.

When simplifying an expression like \(\sec^2 x \sin^2 x\), recognizing that \(\sec^2 x\) is the reciprocal of \(\cos^2 x\) opens the door to further simplification. By substituting \(\sec x\) with \(\frac{1}{\cos x}\), we get \(\frac{\sin^2 x}{\cos^2 x}\), which simplifies further to another trigonometric function.

Sine and Cosine

The sine and cosine functions form the foundation of trigonometry. \(\sin x\) and \(\cos x\) represent the y-coordinate and x-coordinate, respectively, of a point on the unit circle. They describe the ratios of sides in a right-angled triangle, with sine representing the opposite over hypotenuse, and cosine representing adjacent over hypotenuse. These functions have a rich relationship, showcased in the Pythagorean identity among other equations.

Understanding how \(\sin x\) and \(\cos x\) relate to each other, especially through identities like the Pythagorean identity, allows us to accomplish tasks such as simplifying \(\sec^2 x \sin^2 x\) to \(\tan^2 x\), because \(\tan x = \frac{\sin x}{\cos x}\) and squaring both sides gives us the ratio of the squares of sine and cosine. This is also where knowing the reciprocal nature of the secant function is crucial, as \(\sec^2 x \sin^2 x \Rightarrow \(\frac{1}{\cos^2 x}\)\sin^2 x\), which allows the simplification to a concise trigonometric function.